Applied Mathematics & Information Sciences

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An A-semiring has commutative multiplication and the property that every proper ideal B is contained in a prime ideal P, with pB, the intersection of all such prime ideals. In this paper, we define homogeneous ideals and their radicals in a graded semiring R. When B is a proper homogeneous ideal in an A-semiring R, we show that pB is homogeneous whenever pB is a k-ideal.We also give necessary and sufficient conditions that a homogeneous k-ideal P be completely prime (i.e., F 62 P;G 62 P implies FG 62 P) in any graded semiring. Indeed, we may restrict F and G to be homogeneous elements of R.

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